On the Distribution of the Information Density of Gaussian Random Vectors: Explicit Formulas and Tight Approximations

Based on the canonical correlation analysis we derive series representations of the probability density function (PDF) and the cumulative distribution function (CDF) of the information density of arbitrary Gaussian random vectors as well as a general formula to calculate the central moments. Using the general results we give closed-form expressions of the PDF and CDF and explicit formulas of the central moments for important special cases. Furthermore, we derive recurrence formulas and tight approximations of the general series representations, which allow very efficient numerical calculations with an arbitrarily high accuracy as demonstrated with an implementation in PYTHON publicly available on GITLAB. Finally, we discuss the (in)validity of Gaussian approximations of the information density. Index Terms information density, information spectrum, probability density function, cumulative distribution function, central moments, Gaussian random vector, canonical correlation analysis ∗∗ Funded in part by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) as part of Germany’s Excellence Strategy – EXC 2050/1 – Project ID 390696704 – Cluster of Excellence "Centre for Tactile Internet with Human-inthe-Loop" (CeTI) of Technische Universität Dresden. ar X iv :2 10 5. 03 92 5v 3 [ cs .I T ] 4 N ov 2 02 1 HUFFMANN, MITTELBACH: ON THE DISTRIBUTION OF THE INFORMATION DENSITY OF GAUSSIAN RANDOM VECTORS 2